基于插入最小二乘的高维时间序列模型变点推断
Inference on the change point in high dimensional time series models via plug in least squares
摘要：
本文研究了变点参数的插入式最小二乘估计，其中变点参数是次高斯分布或次指数分布下的高维随机向量的均值。我们得到了充分的条件下，该估计具有足够的适应性对插件的平均参数的估计，以产生一个最佳的收敛速度$O_p(ξ^{-2})$的整数规模。在亚高斯和亚指数情况下分别提供$s\log (p\vee T)=o(\surd(Tl_T))$或$s\log^{3/2}(p\vee T)=o(\surd(Tl_T))$的同时，保持该速率，同时允许高维度以及潜在减小的跳跃大小$ξ,$。这里$s,p,T$和$l_T$表示稀疏参数、模型维数、采样周期和变化点与其参数边界的分离。此外，由于收敛速度不受$s,p$和$T,$的对数项的影响，因此允许极限分布的存在。这些分布，然后导出为{ \it argmax}的双边负漂移布朗运动或双边负漂移随机游动下消失和非零跳跃大小制度，分别。从而允许在高维设置中推断变点参数。所提出的方法的实施提供了可行的算法。理论结果与蒙特卡洛模拟支持。
英文摘要：
We study a plug in least squares estimator for the change point parameter where change is in the mean of a high dimensional random vector under subgaussian or subexponential distributions. We obtain sufficient conditions under which this estimator possesses sufficient adaptivity against plug in estimates of mean parameters in order to yield an optimal rate of convergence $O_p(ξ^{-2})$ in the integer scale. This rate is preserved while allowing high dimensionality as well as a potentially diminishing jump size $ξ,$ provided $s\log (p\vee T)=o(\surd(Tl_T))$ or $s\log^{3/2}(p\vee T)=o(\surd(Tl_T))$ in the subgaussian and subexponential cases, respectively. Here $s,p,T$ and $l_T$ represent a sparsity parameter, model dimension, sampling period and the separation of the change point from its parametric boundary. Moreover, since the rate of convergence is free of $s,p$ and logarithmic terms of $T,$ it allows the existence of limiting distributions. These distributions are then derived as the {\it argmax} of a two sided negative drift Brownian motion or a two sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively. Thereby allowing inference of the change point parameter in the high dimensional setting. Feasible algorithms for implementation of the proposed methodology are provided. Theoretical results are supported with monte-carlo simulations.
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